Transcription of 2-D Fourier Transforms - New York University
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2-D Fourier Transforms - New York University
2-D Fourier Transforms Yao Wang Polytechnic University University , Brooklyn Brooklyn, NY 11201. With contribution from Zhu Liu, Onur Guleryuz, and Gonzalez/Woods, Digital Image Processing, 2ed Lecture Outline Continuous Fourier Transform (FT). 1D FT (review). 2D FT. Fourier Transform for Discrete Time Sequence (DTFT). 1D DTFT (review). 2D DTFT. Linear Li C. Convolution l ti 1D, Continuous vs. discrete signals (review). 2D. Filter Design Computer Implementation Yao Wang, NYU-Poly EL5123: Fourier Transform 2. What is a transform? Transforms are decompositions of a function f(x). into some basis functions (x, u). u is typically the freq. index. Yao Wang, NYU-Poly EL5123: Fourier Transform 3. Illustration of Decomposition 3. f 3. f = 1 1+ 2 2+ 3 3. o 2. 2. 1. 1. Yao Wang, NYU-Poly EL5123: Fourier Transform 4. Decomposition Ortho-normal basis function 1, u1 u2. ( x, u1 ) * ( x, u2 )dx 0, u1 u2. Forward . Projection of ) ( x, u ) . F (u ) f ( x), f ( x) . *. ( x, u )dx d f(x) onto (x,u).
2D FT in polar coordinate (r θ)and(ρØ) x rcos , y rsin , u cos , v sin . ... • One Dimensional DTFT – f(n) is a 1D discrete time sequencef(n) is a 1D discrete time sequence – Forward Transform F( ) i i di i ith i d ITf ...
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